Linear Algebra, scalars to find det(A)

[concon]

Homework Statement

Let X1, X2,...,Xn be scalars. Calculate det(A) where
A= nxn matrix with [ x1+1 x1+2...x1+n
x2+1 x2+2...x2+n
... ... ...
xn+1 xn+2...xn+n]

Homework Equations

det(A) = (aij)(-1)^(i+j)det(aij)

The Attempt at a Solution

I have no clue how to even begin solving this problem I tried using above formula to no avail.
Please help!

 

[concon]

Spacing on matrix got messed up so imagine it with the rows lined up correctly

 

[AlephZero]

Review what you know about how determinants, row (or column) operations on the matrix, and linear dependence.

Hint: the answer is very simple, when n > 2.

 

[concon]

Can you explain a little more about what operations you are talking about? I am really confused!

 

[kduna]

Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.

 

[concon]

Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.

How would that help me calculate the determinant when n is unknown?

 

[Ray Vickson]

How would that help me calculate the determinant when n is unknown?

Start with simple cases of, say, n = 2, 3 and maybe 4, just to see what is going on. Then do it for general n.

 

[AlephZero]

How would that help me calculate the determinant when n is unknown?

The answer doesn't depend on n, except that n = 1 and n = 2 are special cases.

I don't know how to give you any more help than "think about row and column operations on the matrix, and linear dependence" without telling you the answer.

Writing out the matrix in full, for n = 3 or n = 4, might help.

If the only thing you know about determinants is your "relevant equation"
det(A) = (aij)(-1)^(i+j)det(aij)
I think you missed a lot of stuff in class, or you haven't read your textbook.

 

[concon]

Have you tried computing the determinant for small n? Such as a 2x2, 3x3, 4x4. I'd recommend doing that.

Yes I did computr for 2 by 2 and 3 by 3, but I do not see a relationship between the determinants. How do I calculate det(A) for nxn?

 

[concon]

Start with simple cases of, say, n = 2, 3 and maybe 4, just to see what is going on. Then do it for general n.

Okay I did determinant calculation for 2x2 and 3x3. How do I calculate for nxn? Please help!

 

[pasmith]

Two hints:
(1) Why is $\det A = B(x_2, \dots, x_n)x_1 + C(x_2, \dots, x_n)$ for some functions $B$ and $C$?
(2) What is the determinant of a matrix with two identical rows?

 

[concon]

Two hints:
(1) Why is $\det A = B(x_2, \dots, x_n)x_1 + C(x_2, \dots, x_n)$ for some functions $B$ and $C$?
(2) What is the determinant of a matrix with two identical rows?

1. I have no clue

2. determinant would be zero. Is the answer zero?

 

[Ray Vickson]

Yes I did computr for 2 by 2 and 3 by 3, but I do not see a relationship between the determinants. How do I calculate det(A) for nxn?

So, what did you get for n = 2 and for n = 3?

 

[concon]

So, what did you get for n = 2 and for n = 3?

For n=2 det(A)= (x1 +1)(x2 +2) - (x1 +2)(x2 +1)
How does that correlate to unknown n?

 

[Ray Vickson]

For n=2 det(A)= (x1 +1)(x2 +2) - (x1 +2)(x2 +1)
How does that correlate to unknown n?

What do you get for n = 3? I mean, expand out everything and simplify it down to as small an expression as you can get. I am 100% serious. Looking at just n = 2 is not enough to reveal the pattern.

 

[AlephZero]

(2) What is the determinant of a matrix with two identical rows?

2. determinant would be zero.

Can you think of a way to do row or column operations on the matrix, to make two rows identical, and not change the determinant?

 

[concon]

Can you think of a way to do row or column operations on the matrix, to make two rows identical, and not change the determinant?

Hey I actually just figured out how to solve it by using row operations and linear dependence and I got zero as the determinant. Is this correct?